# Titles and abstracts

## Structures in Enumerative Geometry

### The University of Sheffield, 23-27 January 2023

## Lectures

**Luca Battistella**:*Alternative compactifications of the moduli space of curves*

The Deligne-Mumford compactification is a smooth orbifold with normal crossings boundary and a projective moduli space. Investigating the birational geometry and intersection theory of the latter has led to many interesting developments in moduli theory in the past fifty years.

I will start with a basic outline of the Hassett-Keel program to study the canonical model of the moduli space by moving along a two-dimensional slice of the pseudo-effective cone. The birational models constructed in the first few steps of this program admit a modular interpretation as they parametrise curves with worst than nodal singularities. Exploring alternative stability conditions is thus (also) a way of improving our understanding of the geometry of the Deligne-Mumford compactification.

I will then focus on small genera: in genus one, the pioneering work of Smyth has recently been enhanced by the application of logarithmic techniques (thanks to Ranganathan, Santos-Parker, Wise, and Bozlee) to produce a host of new compactifications - due to Bozlee, Kuo, and Neff - involving some nice combinatorics, which turn out to exhaust all possible compactifications with Gorenstein singularities and distinct markings.

In genus two, work of Carocci and myself has highlighted the relationship between Gorenstein singularities and (holomorphic, tropical) differentials, again in the unifying framework of logarithmic geometry. I will try to convince you that Gorenstein singularities are a mild generalisation of nodes with a lot of potential applications to the study of linear series, enumerative geometry, etc.

**Francesca Carocci:***Logarithmic and tropical techniques in moduli theory*

The first sentence of Ogus’ book “Lectures in Logarithmic geometry” reads: ‘Logarithmic geometry was developed to deal with two fundamental and related problems in algebraic geometry: compactifications and degenerations’.

At the same time, the combinatorial counterpart of log-geometry, tropical geometry, comes equipped with a convenient “toric toolkit” which has also proved extremely useful in studying several questions regarding moduli spaces and their invariants.

We explore this philosophy in the case of moduli spaces of curves and maps. Abramovich, Cavalieri, Chen, Gross, Marcus, Siebert, Ranganathan, Wise and many others showed us the way.

We will begin by introducing logarithmic stable curves, their moduli spaces and the relation with the Deligne-Mumford compactification. We will then proceed to exploit the newly befriended log- and tropical curves to study both moduli spaces of absolute and log-(expanded) stable maps.

**Jeongseok Oh**:*Counting sheaves on Calabi-Yau 4-folds*

Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We constructed an algebraic virtual cycle. A key step is a localisation of Edidin-Graham's square root Euler class for SO(2n,C) bundles to the zero locus of an isotropic section, or to the supprot of an isotropic cone.

We also develop a theory of complex Kuranishi structures on projective schemes which are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to the moduli spaces to prove the two virual cycles coincide in homology after inverting 2 in the coefficients. In particular, when Borisov-Joyce's real virtual dimension is odd, their virtual cycle is torsion.

This is a joint work with Richard Thomas.

## Research talks

**Alessio Corti**:*C**luster varieties and degenerations of Fano varieties*

I will present a conjecture, sketch a proof in dim = 2, and present some evidence

**Michel van Garrel**:*The Prism of Intrinsic Mirror Symmetry*

In mirror symmetry as for vinyl records, there are two sides, an A-side and a B-side. Unlike vinyl records though, both sides are supposed to be equivalent. This correspondence is usually proven through the computation of each side, which limits the scope of results. Intrinsic Mirror Symmetry by Gross and Siebert changes the game. The full enumerative invariants of the A-side construct the B-side. This is the mirror construction. Then the mirror theorem becomes a prism (period integrals) applied to the B-side in order to recover specific enumerative invariants of the A-side. In joint work with Ruddat and Siebert, we show how this works for log Calabi-Yau varieties with smooth boundary, such as Fanos with smooth anticanonical divisor.

**Dominic Joyce:***The structure of invariants counting coherent sheaves on complex surfaces*

Let X be a complex projective surface with geometric genus p_{g} > 0. We can form moduli spaces M_{(r,a,k)}^{st} ⊂ M_{(r,a,k)}^{ss} of Gieseker (semi)stable coherent sheaves on X with Chern character (r,a,k), where we take the rank r to be positive. In the case in which stable = semistable, there is a (reduced) perfect obstruction theory on M_{(r,a,k)}^{ss}, giving a virtual class [M_{(r,a,k)}^{ss}]_{virt} in homology.

By integrating universal cohomology classes over this virtual class, one can define enumerative invariants counting semistable coherent sheaves on X. These have been studied by many authors, and include Donaldson invariants, K-theoretic Donaldson invariants, Segre and Verlinde invariants, part of Vafa-Witten invariants, and so on.

In my paper https://arxiv.org/abs/2111.04694, in a more general context, I extended the definition of the virtual class [M_{(r,a,k)}^{ss}]_{virt} to allow strictly semistables, proved wall-crossing formulae for these classes and associated “pair invariants”, and gave an algorithm to compute the invariants [M_{(r,a,k)}^{ss}]_{virt} by induction on the rank r, starting from data in rank 1, which is the Seiberg-Witten invariants of X and fundamental classes of Hilbert schemes of points on X. This is an algebro-geometric version of the construction of Donaldson invariants from Seiberg-Witten invariants; it builds on work of Mochizuki 2008.

This talk will report on a project to implement this algorithm, and actually compute the invariants [M_{(r,a,k)}^{ss}]_{virt} for all ranks r > 0. I prove that the [M_{(r,a,k)}^{ss}]_{virt} for fixed r and all a,k with a fixed mod r can be encoded in a generating function involving the Seiberg-Witten invariants and universal functions in infinitely many variables. I will spend most of the talk explaining the structure of this generating function, and what we can say about the universal functions, the Galois theory and algebraic numbers involved, and so on. This proves several conjectures in the literature by Lothar Göttsche, Martijn Kool, and others, and tells us, for example, the structure of U(r) and SU(r) Donaldson invariants of surfaces with b^{2}_{+} > 1 for any rank r ≥ 2.

**David Kern**:*Derived geometric CohFT for orbifold quasimap theory*

The Gromov–Witten invariants of a smooth projective variety produce a Cohomological Field Theory, a certain algebraic structure controlled by the homologies of the moduli stacks of stable curves. Mann–Robalo showed that, using derived geometry, it can be lifted from the cohomological setting to the geometric one.

When the target is a stack, it is known from Abramovich–Graber–Vistoli that the CohFT is only exhibited on (a “cyclotomic” decomposition of) its inertia stack. I will explain how the orbifold structure of the target can be used to extend the GW stability condition to the family of quasimap theories on it, and how Mann–Robalo’s construction adapts to a geometric CohFT in which the cyclotomic inertia appears naturally.

**Navid Nabijou**:*Roots and logs in the enumerative forest*

Logarithmic and orbifold structures provide two different paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under strata blowups of the target, while the orbifold theory is not. By identifying a suitable system of blowups, we define a “limit" orbifold theory and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood. Time permitting, I will discuss related upcoming work (with the same coauthors), in which we apply techniques from toroidal intersection theory to the study of invariants with negative tangency orders.

**Nicola Pagani**:*A wall-crossing formula for universal Brill-Noether classes*

We will discuss an explicit graph formula, in terms of boundary strata classes, for the wall-crossing of universal (=over the moduli space of stable curves) Brill-Noether classes. More precisely, fix two stability conditions for universal compactified Jacobians that are on different sides of a wall in the stability space. Then we can compare the two universal Brill-Noether classes on the two compactified Jacobians by pulling one of them back along the (rational) identity map. The calculation involves constructing a resolution by means of subsequent blow-ups. If time permits, we will discuss the significance of our formula and potential applications. This is joint with Alex Abreu.

**Renata****Picciotto**:*The derived moduli of sections and virtual pushforwards*

Derived algebraic geometry provides a powerful set of tools to enumerative geometers, giving geometric spaces which encode the "virtual structures" of the moduli problems . I will discuss a joint work with D. Kern, E. Mann and C. Manolache in which we define a derived enhancement for the moduli space of sections. This enriched space neatly encodes the perfect obstruction theory and virtual structure sheaves of many theories. Special cases include Gromov-Witten and quasimaps theories. To illustrate the potential of this approach, I will explain how we use local derived charts to prove a virtual pushforward formula between stable maps and quasimaps without relying on torus localization.

**Dhruv Ranganathan**:*Logarithmic Gromov-Witten theory and the double ramification cycle*

The logarithmic Gromov-Witten theory of the pair (X,D) where X is toric and D is the toric boundary is probably the most basic target geometry in the subject and, in retrospect, was hiding behind work of Mikhalkin and Nishinou-Siebert in the early 2000s on tropical correspondence. I will explain how these logarithmic GW invariants can be expressed as products of natural tautological classes and double ramification cycles in the “logarithmic” tautological ring of the moduli space of curves. Practically carrying out calculations leads naturally to tropical geometry and the correspondence theorems. The results rely on a pleasant mix of ingredients: logarithmic birational invariance, product formulas, strict transform formulas in intersection theory, and a curious gadget called the logarithmic algebraic torus. I’ll try to give a sense for how these ideas fit together, and the broader context for the results. This is joint work with Sam Molcho (ETH) and Ajith Urundolil Kumaran (Cambridge).

**Richard Thomas**:*Counting sheaves by counting curves*

DT invariants count stable bundles and sheaves on a Calabi-Yau 3-fold X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants counting ideal sheaves of curves in X.

By the MNOP conjecture the latter invariants are determined by the Gromov-Witten invariants of X. Along the way we also show all these invariants are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.